Type Theory folks formed two groups; one stands for “proofs are not important”,
one stands for “proofs are important”.
Someone claims that the identity proof between two objects is unique – which
is related to the K rule.
The other people believe that the proof of identity is significant – the same
theorem can
have different proofs, where the identity relation on these proofs are also
nontrivial and can be proved to be equivalent or not.

Cubical model

In the Cubical model, there’s no pattern matching because you can never tell
how many proofs there can be (so no case analysis).
Instead, there are interval operations that can construct proofs via existing proofs
in a more detailed way.